My notation is rusty, but am I right in thinking Sn is the group of all permutations of n elements, while An is the group of all even permutations of n elements?

If a subgroup G of Sn is not contained in An, this means it contains at least one odd element. Let E be such an element. Then consider the mapping f: G->G which maps x to Ex. This is a bijection (by group properties) and it maps every odd element to an even element and vice versa. Hence, there must be the same number of odd and even elements.